C*-algebra generated by four projections with sum equal to 2

We describe the C*-algebra generated by four orthogonal projections p₁,p₂,p₃,p₄, satisfying the linear relation p₁ + p₂ + p₃ + p₄ = 2I. The simplest realization by 2 × 2-matrixfunctions over the sphere S² is given.

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Date:	2004
Main Author:	Savchuk, Y.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2004
Series:	Algebra and Discrete Mathematics
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Cite this:	C*-algebra generated by four projections with sum equal to 2 / Y. Savchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 126–134. — Бібліогр.: 8 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1565762025-02-23T17:47:29Z C-algebra generated by four projections with sum equal to 2 Savchuk, Y. We describe the C-algebra generated by four orthogonal projections p₁,p₂,p₃,p₄, satisfying the linear relation p₁ + p₂ + p₃ + p₄ = 2I. The simplest realization by 2 × 2-matrixfunctions over the sphere S² is given. Author expresses his gratitude to Prof. Yuri˘ı S. Samo˘ılenko for the problem stating and permanently attention. I am also indebted to Dr. Daniil P. Proskurin for his critical remarks and helpful discussions. 2004 Article C*-algebra generated by four projections with sum equal to 2 / Y. Savchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 126–134. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 46L05, 47L50. https://nasplib.isofts.kiev.ua/handle/123456789/156576 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We describe the C*-algebra generated by four orthogonal projections p₁,p₂,p₃,p₄, satisfying the linear relation p₁ + p₂ + p₃ + p₄ = 2I. The simplest realization by 2 × 2-matrixfunctions over the sphere S² is given.
format	Article
author	Savchuk, Y.
spellingShingle	Savchuk, Y. C*-algebra generated by four projections with sum equal to 2 Algebra and Discrete Mathematics
author_facet	Savchuk, Y.
author_sort	Savchuk, Y.
title	C*-algebra generated by four projections with sum equal to 2
title_short	C*-algebra generated by four projections with sum equal to 2
title_full	C*-algebra generated by four projections with sum equal to 2
title_fullStr	C*-algebra generated by four projections with sum equal to 2
title_full_unstemmed	C*-algebra generated by four projections with sum equal to 2
title_sort	c*-algebra generated by four projections with sum equal to 2
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2004
url	https://nasplib.isofts.kiev.ua/handle/123456789/156576
citation_txt	C*-algebra generated by four projections with sum equal to 2 / Y. Savchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 126–134. — Бібліогр.: 8 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT savchuky calgebrageneratedbyfourprojectionswithsumequalto2
first_indexed	2025-11-24T04:43:53Z
last_indexed	2025-11-24T04:43:53Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2004). pp. 126 – 134 c© Journal “Algebra and Discrete Mathematics” C ∗-algebra generated by four projections with sum equal to 2 Yuri Savchuk Communicated by L. Turowska Abstract. We describe the C∗-algebra generated by four orthogonal projections p1, p2, p3, p4, satisfying the linear relation p1 + p2 + p3 + p4 = 2I. The simplest realization by 2 × 2-matrix- functions over the sphere S2 is given. Introduction In the present paper we consider a realization of a certain C∗-algebra A with irreducible representations of dimensions equal to 1 or 2 only, as a C∗-algebra of continuous matrix-functions over S2 with boundary conditions. C∗-algebras with restriction on the dimensions of the irreducible rep- resentations are the object of intensive investigations, started from the works of Gelfand-Naimark, Fell, Tomiyama-Takesaki, Vasil’ev (see [4], [6], [7]). An interesting fact is that the property for a C∗-algebra A to have irreducible representations of dimensions less or equal to n can be formu- lated in pure algebraic way. Let Fn denote the following polynomial of degree n in n non-commuting variables: Fn(x1, x2, . . . , xn) = ∑ σ∈Sn (−1)p(σ)xσ(1) . . . xσ(n), 2000 Mathematics Subject Classification: 46L05, 47L50. Key words and phrases: matrix-functions, projections, finitely generated C ∗- algebras. Jo u rn al A lg eb ra D is cr et e M at h .Y. Savchuk 127 where Sn is the symmetric group of degree n, p(σ) is the parity of a permutation σ ∈ Sn. We say that an algebra A is an algebra with Fn identity if for all x1, . . . , xn ∈ A, we have Fn(x1, . . . xn) = 0. The Amitsur-Levitsky theorem says that the matrix algebra Mn(C) is an al- gebra with F2n identity. A C∗-algebra A has irreducible representations of dimension less or equal to n iff A satisfies the F2n condition (see [5]). One of the basic C∗-algebra classes with F2n identity is the class of n- homogeneous algebras. Recall, that an algebra is called n-homogeneous iff all its irreducible representations are of dimension n. Any n-homoge- neous C∗-algebra can be described in terms of algebraic bundles, see [6] or [7]. It is also convenient to realize these algebras as algebras of continuous matrix-functions. For example, it was proved in [1], that one has exactly n pairwise non-isomorphic n-homogeneous C∗-algebras having the dual space S2 (see [1]). We will denote them by An,k, k = 0, n− 1. Such algebras can be realized in the following way. Let S1 = {z ∈ C : \|z\| = 1} be the boundary of the unit disk D2 in the complex plane and consider Vk : S1 −→ U(n), z 7→ diag(zk, 1, . . . , 1), k = 0, n− 1. Then An,k = {f ∈ C(D2 −→M2(C))\|f(z) = Vk(z) ∗f(1)Vk(z), z ∈ S1}. Evidently, the dual space is homeomorphic to D2/S1 ' S2 (see [1] for more details). An analogous realization of n-homogeneous algebra, having the two- dimensional torus as the dual space, was presented in [2]. Namely, any such algebra is isomorphic to BV,W = {g ∈ C([0, 1]2 −→Mn(C))\|g(0, s) = V ∗g(1, s)V, g(t, 0) = W ∗g(t, 1)W, s, t ∈ [0, 1]}, where V,W ∈ U(n) are some unitary matrices such that VWV ∗W ∗ is a scalar matrix. Note, that concrete finitely generated F2n-algebras are mostly non- homogeneous. Indeed, the group C∗-algebra of any non-commutative finite group satisfies the F2n condition for some n, but it is not homo- geneous. The group algebra of G = Z2 ∗ Z2 gives an example of F4 algebra corresponding to infinite discrete group. One can also generate C∗(Z2 ∗Z2) by the free pair of projections. Indeed, it is easy to see, that C∗(Z2 ∗ Z2) = C∗〈p1, p2\|p2 k = pk = p∗k, k = 1, 2〉 := P2. A realization of P2 as algebra of matrix-functions was constructed in [8]. Jo u rn al A lg eb ra D is cr et e M at h .128 C∗ -algebra generated by four projections Namely, P2 = {f ∈ C([0, 1] −→M2(C))\|f(0), f(1) are diagonal}. In this paper we study the C-algebra A generated by four projections (self-adjoint idempotents) P1, P2, P3, P4 satisfying the following relation: P1 + P2 + P3 + P4 = 2I. The algebra A is an enveloping of the -algebra: Ã = C 〈 Pi \| 4∑ i=1 Pi = 2I, Pi = P ∗ i = P 2 i , i = 1 . . . 4 〉 . In Theorem 1, we realize A as an algebra of continuous 2 × 2 matrix- functions with some boundary conditions. In the theorem 2 we give the most simple of possible realizations of A. 1. Preliminaries In this Section, for convenience of the reader, we recall some information used below. Definition 1. Let A be a -algebra, having at least one representation. Then a pair (A, ρ) of a C∗-algebra A and a homomorphism ρ : A −→ A is called an enveloping pair for A if every irreducible representation π:A −→ B(H) factors uniquely through the A, i.e. there is precisely one irreducible representation π1 of algebra A satisfying π1 ◦ ρ = π. The algebra A is called an enveloping for A. The following statement is a simple corollary of the noncommutative analogue of the Stone-Weierstrass theorem for C∗-algebras (see Glimm- Stone-Weierstrass theorem in [4] or [7]). Statement 1. Let Y be a compact Hausdorff space. Let C ⊆ B be sub- algebras of A = C(Y −→ Mn(C)). For every pair x1, x2 ∈ Y define A(x1, x2) (B(x1, x2), C(x1, x2) respectively) as: A(x1, x2) := {(f(x1), f(x2)) ∈Mn(C) ×Mn(C)\|f ∈ A (f ∈ B, f ∈ C respectively)}. Then B = C ⇐⇒ B(y1, y2) = C(y1, y2) ∀y1, y2 ∈ Y. Jo u rn al A lg eb ra D is cr et e M at h .Y. Savchuk 129 In the next section we will also need a classification of all irreducible representation of Ã (see [5] for more details). Namely, irreducible rep- resentations are either 1-dimensional or 2-dimensional. The images of generators of Ã in two-dimensional representations have the following form : P1(a, b, c) = 1 2 ( 1 + a −b− ic −b+ ic 1 − a ) , P2(a, b, c) = 1 2 ( 1 − a b− ic b+ ic 1 + a ) , P3(a, b, c) = 1 2 ( 1 − a −b+ ic −b− ic 1 + a ) , P4(a, b, c) = 1 2 ( 1 + a b+ ic b− ic 1 − a ) . where a2 +b2 +c2 = 1 and the space of parameters (a, b, c) corresponding to irreducible pairwise non-equivalent 2-dimensional representations is (a part of the unit sphere in R 3): P = {(a, b, c)\|a > 0, b > 0, c ∈ R} ⋃ {(a, b, c)\|a = 0, b > 0, c > 0} ⋃ ⋃ {(a, b, c)\|a > 0, b = 0, c > 0}. Note that when (a, b, c) ∈ {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, the formulas for Pk give reducible representations of Ã, moreover, any one-dimensional representation of Ã can be obtained by decomposition of some of these reducible ones on irreducible components. We will denote by P the closure of P in R 3. Evidently P = {(a, b, c)\|a2 + b2 + c2 = 1, a ≥ 0, b ≥ 0}. 2. The structure of enveloping C ∗-algebra In this section we give a description of the enveloping C∗-algebra A of Ã. Theorem 1 realizes A by matrix-functions, and Theorem 2 gives the simplest of all descriptions for A. Theorem 1. Let X = {(x, y)\|(x, y) ∈ R 2, \|x\| + \|y\| ≤ 1}, V = ( 1 0 0 −1 ) , W = ( 0 1 1 0 ) , A0 = {f ∈ C(X −→M2(C))\|f(t, 1 − t) = V f(−t, 1 − t)V, f(t, t− 1) = Wf(−t, t− 1)W, t ∈ [0, 1]}, then A ' A0. Proof. Consider the functions Pi = Pi(a, b, c), i = 1, 4 naturally corre- sponding to generators Pi defined on P . Let Â ⊆ C(P −→ M2(C)) be Jo u rn al A lg eb ra D is cr et e M at h .130 C∗ -algebra generated by four projections the C∗-algebra generated by Pi. It is easy to check, that Â is an envelop- ing C∗-algebra of Ã, i.e. A. Indeed, we have homomorphism of Ã into Â, which satisfies the universal property, so Â is enveloping algebra by Definition 1. We will show, that Â coincides with A = {f ∈ C(P →M2(C))\| V f(s, 0, t)V = f(s, 0,−t), Wf(0, s, t)W = f(0, s,−t), s2 + t2 = 1}. To do so we apply Statement 1. Let us check that Â ⊆ A. Indeed, it is easy to check, that Pi satisfy the boundary conditions from the definition of A, so we have Pi ∈ A. The fact, that P is space of pairwise non-equivalent irreducible rep- resentations insures that: Â(x1, x2) = A(x1, x2) = M2(C) ×M2(C), ∀x1, x2 ∈ P, and automatically: Â(x1, x2) = A(x1, x2) ⊂M2(C) ×M2(C), ∀x1, x2 ∈ P . So, by Statement 1 we have Â = A. Choose a homeomorphism between P and X which maps the points (1, 0, 0), (0, 0,±1), (0, 1, 0) ∈ P to the points (0, 1), (±1, 0), (0,−1) ∈ X, correspondingly. This homeomorphism induces the isomorphism between A and A0. Remark. It is easy to show that this theorem implies that the space of primitive ideals of algebra A is the same as for algebra of all continuous matrix-functions on the sphere S2 having values in diagonal matrix in three fixed points. It turns out that A is isomorphic to such an algebra. Theorem 2. Let B = {f ∈ C(S2 −→M2(C))\|f(xi) ∈ Bi 'M1(C) ⊕M1(C), i = 1, 2, 3}, where x1, x2, x3 are fixed points of the sphere S2, then A ' B. Proof. We will prove this theorem in a few steps, sequently building dif- ferent realizations of A. I. It is easy to see, that the algebra A0 is isomorphic to the algebra A1, where A1 = {(f1, f2)\|f1, f2 ∈ C(X1 −→M2(C)), f1(s, 0) = f2(s, 0), s ∈ [−1, 1], V f1(t, 1 − t)V = f1(−t, 1 − t), Wf2(t, 1 − t)W = f2(−t, 1 − t), t ∈ [0, 1]}, X1 = {(x, y)\|(x, y) ∈ R 2, \|x\| + y ≤ 1, y ≥ 0} Jo u rn al A lg eb ra D is cr et e M at h .Y. Savchuk 131 (the norm on the algebra A1 is natural: ‖ (f1, f2) ‖= max(‖ f1 ‖, ‖ f2 ‖)). The boundary conditions for A1 imply that: f1(0, 1) ∈ {( a 0 0 b )} a,b∈C , f2(0, 1) ∈ {( c d d c )} c,d∈C , f1(1, 0) = f2(1, 0) = V f1(−1, 0)V = Wf2(−1, 0)W ∈ {( e f −f e )} e,f∈C . Let R1 = 1√ 2 ( 1 −1 1 1 ) , R2 = 1√ 2 ( 1 1 i −i ) . One can check, that R∗ 1 ( c d d c ) R1, R ∗ 2 ( e f −f e ) R2 are diagonal ma- trices for any c, d, e, f ∈ C. So, one has natural isomorphism, which will be used in considerations below. {( c d d c )} c,d∈C 'M1(C)⊕M1(C), {( e f −f e )} e,f∈C 'M1(C)⊕M1(C). II. Let λ1 : [0, 1] −→ U(2), t 7→ ( 1 0 0 eiπt ) , λ2 : [0, 1] −→ U(2), t 7→ ei πt 2 ( cosπt 2 −isinπt 2 −isinπt 2 cosπt 2 ) be homotopies joining the unit matrix E with V and W respectively. Construct maps µi : X1 −→ U(2) by the rule: (x, y) 7→ λi ( x+ 1 − y 2(1 − y) ) , (x, y) 6= (0, 1), (0, 1) 7→ E. Neither µ1 nor µ2 is continuous, nevertheless it is easy to check, that ∀(f1, f2) ∈ A1, (µ∗1f1µ1, µ ∗ 2f2µ2) is a pair of continuous matrix-functions (here µ∗i (x), x ∈ X1, means the adjoint of the matrix µi(x)). The corre- spondence: A1 3 (f1, f2) 7→ (µ∗1f1µ1, µ ∗ 2f2µ2), induces an isomorphism: A1 ' A2 = {(µ∗1f1µ1, µ ∗ 2f2µ2)\|(f1, f2) ∈ A1} = = {(g1, g2)\| g1, g2 ∈ C(X1 −→M2(C)), gi(0, 1) ∈ A (i) 2 'M1(C) ⊕M1(C), g1(t, 1 − t) = g1(−t, 1 − t), g2(t, 1 − t) = g2(−t, 1 − t), t ∈ [0, 1], λ1((s+ 1)/2)g1(s, 0)λ∗1((s+ 1)/2) = = λ2((s+ 1)/2)g2(s, 0)λ∗2((s+ 1)/2), s ∈ [−1, 1], }. Jo u rn al A lg eb ra D is cr et e M at h .132 C∗ -algebra generated by four projections III. Further, the boundary conditions g1(t, 1 − t) = g1(−t, 1 − t), g2(t, 1 − t) = g2(−t, 1 − t), t ∈ [0, 1] for algebra A2 allow us to replace X1 by X1/ ∼ where the equivalence relation ∼ is defined as follows: (t, 1 − t) ∼ (−t, 1 − t), t ∈ [0, 1], and we can consider the algebra A2 as an algebra of pairs of functions on the quotient space X1/ ∼. Evidently X1/ ∼ is homeomorphic to the closed unit disk D2 in R 2. We denote this disk by X2. In the following, it will be convenient for us to consider X2 as the unit disk with center (0, 1). In the polar coordinates one has: X2 = {(rcosφ, rsinφ) ∈ R 2\| r ≤ 2sinφ, 0 ≤ φ ≤ π}. Below, for any x ∈ X, by [x] we denote its class in X1/ ∼. We can suppose that the homeomorphism ψ : X1/ ∼ −→ X2 maps [(0, 1)] to the center of disk and the image of [−1, 1] × {0} is the boundary of D2 ∂X2 = {(rcosφ, rsinφ) ∈ R 2\|r = 2sinφ, 0 ≤ φ ≤ π}. To be more precise, one can choose ψ such that: [(0, 1)] 7→ (0, 1) ∈ D2, [(s, 0)] 7→ (2 sin(π(s+ 1)/2), π(s+ 1)/2) ∈ ∂D2, s ∈ [−1, 1]. The explanations given above show that one can consider the elements of A2 as the functions on the quotient space. So one has the isomorphism: A2 ' A3 = {(h1, h2) = (h1(r, φ), h2(r, φ))\|hi ∈ C(X2 −→M2(C)), hi(1, π/2) ∈ A (i) 3 'M1(C) ⊕M1(C), h1(2sinφ, φ) = λ∗1 ( φ π ) λ2 ( φ π ) h2(2sinφ, φ)λ∗2 ( φ π ) λ1 ( φ π ) , φ ∈ [0, π]}. The boundary conditions in the point (0, 0) imply that h1(0, 0) = h2(0, 0) ∈ {( e f −f e )} e,f∈C . IV. To prove an isomorphism A ' B we construct a map (non- continuous!): ν = ν(r, φ) : X2 −→M2(C), (r, φ) 7→ ( e i rφ 4sinφ cosφ 2 e i r(φ−π) 4sinφ sinφ 2 −e−i r(φ−π) 4sinφ sinφ 2 e −i rφ 4sinφ cosφ 2 ) , r 6= 0, (0, 0) 7→ ( 1 0 0 1 ) . Jo u rn al A lg eb ra D is cr et e M at h .Y. Savchuk 133 Note, that the restriction of ν on the set {(2sinφ, φ)\|φ ∈ [0, π]} = ∂X2 coincides with λ∗1 ( φ π ) λ2 ( φ π ) = ( ei φ 2 cosφ 2 −iei φ 2 sinφ 2 −ie−i φ 2 sinφ 2 e−i φ 2 cosφ 2 ) = = ( ei φ 2 cosφ 2 ei φ−π 2 sinφ 2 −e−i φ−π 2 sinφ 2 e−i φ 2 cosφ 2 ) . One can check that for every pair (h1, h2) ∈ A3, (h1, νh2ν ∗) is also a pair of continuous matrix function, so the correspondence: A3 3 (h1, h2) 7→ (h1, νh2ν ∗). induces an isomorphism: A3 ' A4 = {(h1, νh2ν ∗)\|(h1, h2) ∈ A3} = = {(k1, k2)\| k1, k2 ∈ C(X2 −→M2(C)); k1(0, 0) = = k2(0, 0) ∈ {( e f −f e )} e,f∈C , ki(1, π/2) ∈ A (i) 4 'M1(C) ⊕M1(C); k1(2sinφ, φ) = k2(2sinφ, φ), φ ∈ [0, π]}. V. The last two conditions for the algebra A4 allow us to unite pairs of functions in one. Namely, let S2 = S2 + ∪ S2 −, where S2 is a unit sphere in R 3, S2 +, S 2 − are the upper and the lower closed half-spheres and χ± : X2 −→ S2 ± be homeomorphisms such that χ+(2sinφ, φ) = χ−(2sinφ, φ), φ ∈ [0, π). Denote by x1, x2, x3 ∈ S2 the points χ+(1, π/2), χ−(1, π/2), χ+(0, 0) = χ−(0, 0). The pair of homeomorphisms χ± defines an isomorphism: A4 ' A5 = {l ∈ C(S2 −→M2(C))\| l(xi) ∈ A (i) 5 'M1(C) ⊕M1(C), i = 1, 2, 3}, given by the rule: A4 3 (k1(x), k2(x)) 7→ l(x) = { k1(χ −1 + (x)), x ∈ S2 +, k2(χ −1 − (x)), x ∈ S2 −. Evidently A5 ' B. The proof is completed. Jo u rn al A lg eb ra D is cr et e M at h .134 C∗ -algebra generated by four projections Acknowledgements. Author expresses his gratitude to Prof. Yurĭı S. Samŏılenko for the prob- lem stating and permanently attention. I am also indebted to Dr. Daniil P. Proskurin for his critical remarks and helpful discussions. References [1] Antonevich A., Krupnik N.: On trivial and non-trivial n-homogeneous C ∗- algebras. Integr.equ.oper.theory 38(2000), 172–189. [2] Disney S., Raeburn I.:Homogeneous C ∗-algebras whose spectra are tori. J.Austral.Math.Soc. (Series A)38(1985), 9–39. [3] Dixmier J.: Les C ∗-algebres et leurs representations.Paris, Gauthier-Villars Edi- teur (1969). [4] Fell J.M.G.:The structure of algebras of operator fields, Acta Math., 106, n. 3-4, (1961), 233–280. [5] Ostrovsky̆ı, V. and Samŏılenko, Yu.: Introduction to the Theory of Representa- tions of Finitely Presented -Algebras. I. Representations by bounded operators. The Gordon and Breach Publishing group, London (1999). [6] Tomiyama J., Takesaki M.: Application of fibre bundles to certain class of C ∗- algbras. Tohoku Math. Journ.13 n.3, (1961), 498–522. [7] Vasil’ev N.: C ∗-algebras with finite dimensional irreducible representations. Us- pehi mat. nauk. XXI., n.1, (1966), 136–154. [8] Vasilevski N., Spitkovski I.: On algebra generated by two projections, Dokl. Akad. Nauk Ukr. SSR, Ser. A 8,(1981),10–13, (Russian). Contact information Y. Savchuk Department of Computer Science, Kyiv Taras Shevchenko University, 64 Volodymyrska st., 01033 Kyiv Ukraine E-Mail: yurius@univ.kiev.ua Received by the editors: 20.04.2004 and final form in 01.10.2004.

C*-algebra generated by four projections with sum equal to 2

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