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Register a new userThe spectra of digraphs with Morita equivalent $C^ast$-algebras
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Eilers et al. have recently completed the geometric classification of unital graph $C^ast$-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the nonzero elements of the spectrum of a finite digraph, which in this paper is allowed to have loops and parallel edges. We consider several different digraph spectra that have been studied in the literature, answering this question for the Laplace and adjacency spectra, their skew counterparts, the symmetric adjacency spectrum, the adjacency spectrum of the line digraph, the Hermitian adjacency spectrum, and the normalized Laplacian, considering in most cases two ways that these spectra can be defined in the presence of parallel edges. We show that the adjacency spectra of the digraph and line digraph are preserved by a subset of the moves, and the skew adjacency and Laplace spectra are preserved by the Cuntz splice. We give counterexamples to show that the other spectra are not preserved by the remaining moves. The same results hold if one restricts to the class of strongly connected digraphs.
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This paper introduces the notion of Rota-Baxter $C^{ast}$-algebras. Here a Rota-Baxter $C^{ast}$-algebra is a $C^{ast}$-algebra with a Rota-Baxter operator. Symmetric Rota-Baxter operators, as special cases of Rota-Baxter operators on $C^{ast}$-algebra, are defined and studied. A theorem of Rota-Baxter operators on concrete $C^{ast}$-algebras is given, deriving the relationship between two kinds of Rota-Baxter algebras. As a corollary, some connection between $ast$-representations and Rota-Baxter operators is given. The notion of representations of Rota-Baxter $C^{ast}$-algebras are constructed, and a theorem of representations of direct sums of Rota-Baxter representations is derived. Finally using Rota-Baxter operators, the notion of quasidiagonal operators on $C^{ast}$-algebra is reconstructed.
The set of distinct eigenvalues of a signed digraph $S$ together with their multiplicities is called its spectrum. The energy of a signed digraph $S$ with eigenvalues $z_1,z_2,cdots,z_n$ is defined as $E(S)=sum_{j=1}^{n}|Re z_j|$, where $Re z_j $ denotes real part of complex number $z_j$. In this paper, we show that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle of length $equiv 0pmod 4$ negative and each cycle of length $equiv 2pmod 4$ positive is of the form $$phi_S(z)=z^n+sumlimits_{j=1}^{lfloor{frac{n}{2}}rfloor}(-1)^j c_{2j}(S)z^{n-2j},$$ where $c_{2j}(S)$ are nonnegative integers. We define a quasi-order relation in this case and show energy is increasing. It is shown that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle negative has the form $$phi_S(z)=z^n+sumlimits_{j=1}^{lfloor{frac{n}{2}}rfloor}c_{2j}(S)z^{n-2j},$$ where $c_{2j}(S)$ are nonnegative integers. We study integral, real, Gaussian signed digraphs and quasi-cospectral digraphs and show for each positive integer $nge 4$ there exists a family of $n$ cospectral, non symmetric, strongly connected, integral, real, Gaussian signed digraphs (non cycle balanced) and quasi-cospectral digraphs of order $4^n$. We obtain a new family of pairs of equienergetic strongly connected signed digraphs and answer to open problem $(2)$ posed in Pirzada and Mushtaq, Energy of signed digraphs, Discrete Applied Mathematics 169 (2014) 195-205.
The program of matrix product states on the infinite tensor product ${mathcal A}^{otimes mathbb Z}$, initiated by Fannes, Nachtergaele and Werner in their seminal paper Commun. Math. Phys. Vol. 144, 443-490 (1992), is re-assessed in a context where $mathcal A$ is an infinite nuclear $C^ast$-algebra. While this setting presents new technical challenges, fine advances on ordered spaces by Kavruk, Paulsen, Todorov and Tomforde enabled us to push through most of the program and to demonstrate that the matrix product states accept generalizations as operator product states.
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $ell(B(I,G))$ and $ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of $A$ is the set of equivalence classes of irreducible $A$-modules. For any finite type $k$-algebra $A$, the spectrum of $A$ is in bijection with the set of primitive ideals of $A$. The stratified equivalence relation preserves the spectrum of $A$ and also preserves the periodic cyclic homology of $A$. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to extended affine Weyl groups.
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