Sequences of operator algebras converging to odd spheres in the quantum Gromov-Hausdorff distance

Marc Rieffel had introduced the notion of the quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on $2$-sphere in this distance, that one finds in many scattered places in the theoretical physics literature. The compact quantum metric spaces and convergence in the quantum Gromov-Hausdorff distance has been explored by a lot of mathematicians in the last two decades. In this paper, we have defined compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman spaces and prove that it converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance

Linear maps preserving parallel matrix pairs with respect to the Ky-Fan kk-norm

Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\|\cdot\|$ if $\|A+\mu B\| = \|A\| + \|B\|$ for some scalar $\mu$ with $|\mu| = 1$. Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan $k$-norms.Comment: 20 page

Non-linear classification of finite-dimensional simple C∗C^*-algebras

A Banach space characterization of simple real or complex $C^*$-algebras is given which even characterizes the underlying field. As an application, it is shown that if $\mathfrak A_1$ and $\mathfrak A_2$ are Birkhoff-James isomorphic simple $C^*$-algebras over the fields $\mathbb F_1$ and $\mathbb F_2$, respectively and if $\mathfrak A_1$ is finite-dimensional with dimension greater than one, then $\mathbb F_1=\mathbb F_2$ and $\mathfrak A_1$ and $\mathfrak A_2$ are (isometrically) $\ast$-isomorphic $C^*$-algebras.Comment: 13 page

A distance formula for tuples of operators

For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and $\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well.Comment: to appear in LA

Prevalence of tooth size discrepancy among North Indian orthodontic patients

Objective: To determine the prevalence of tooth size discrepancy (TSD) in a representative orthodontics population, to explore how many millimeters of TSD is clinically significant and to determine the ability of simple visual inspection to detect such a discrepancy. Materials and Methods: The sample comprised 150 pretreatment study casts with fully erupted and complete permanent dentitions from first molar to first molar, which were selected randomly from records of the orthodontic patients. The mesiodistal diameters of the teeth were measured at contact points using digital calipers and the Bolton′s analysis was carried out on them. Simple visual estimation of Bolton discrepancy was also performed. Results: In the sample group, 24% of the patients had anterior tooth width ratios and 8% had total arch ratios greater than ±2 standard deviation (SD) from Bolton′s means. For the anterior analysis, correction greater than ±2 mm was required for 24% of patients in the upper arch or 14% in the lower arch. For the total arch analysis, correction greater than ±2 mm was required for 36% of patients in the upper arch or 32% in the lower arch. Conclusion: Bolton′s analysis should be routinely performed in all orthodontic patients and the findings should be included in orthodontic treatment planning. 2 mm of the required tooth size correction is an appropriate threshold for clinical significance. Visual estimation of TSD has low sensitivity and specificity. Careful measurement is more frequently required in clinical practice than visual estimation would suggest